Turn Obj-sequence into FBX file for Unity?
You can try Import OBJ Sequence As Shape Keys blender add-on from here. As described by you and shown in the video on the add-on page, it seems to work in your case.

Download the add-on from here (Windows).

Initial setup is as follows:

Select “File” - “Import” - “Wavefront.obj” from the “file” menu and navigate to the directory which holds the OBJ sequence
Select the first OBJ file in the sequence and click “Import OBJ”.
Select the newly imported object in the 3D view or the outliner
Select the “Object Data” tab in the main properties panel
In the “Shape keys” section, click the “Add shape key to object” button (+)
Uncheck the “Relative” Checkbox and set the interpolation dropdown menu to “Linear”
Change the current frame in the timeline to “0” and set a keyframe for “Evaluation time” (as “0”)
Change the current frame on the timeline to “1000”
Enter “10000” as evaluation time. (it seems to go in steps of 10 strangely!), then set a keyframe for “Evaluation time”
Change to the graph editor window, then select the curve for “Evaluation time”
Select “Key” - “Interpolation Mode” - “Linear” from the graph editor menu.
Change back to the 3D view.


Once you done that, you use the script to import the sequence:

Select “File” - “Import” - “OBJ as Shapekey” from the “file” menu and navigate to the directory which holds the OBJ sequence.
Select the rest of the sequence(all except for the first obj file in the sequence) and click “Import OBJ as Shape Keys”.
Wait as it loads the sequence…


A video tutorial..